(Warning: lengthy philosophical discourse. A bit of formal stuff
mentioned in the penultimate paragraph.)
Dean Buckner has challenged us to try to think what Cantor's diagonal
argument proves. In particular, what the diagonal argument showing that
(*) the natural numbers cannot be put in one-to-one
correspondence with the infinite sequences of
digits
proves-- so the challenge is to explain what (*) means. The happy
Platonist will be content to say that (*) means what it says, that there is
no -one-one correspondence between the natural numbers and those infinite
sequences of digits that (in Plato's heaven) exist. Dean's challenge,
then, is to say something that will seem illuminating to the non-Platonist.
[[In what follows I will take some liberties with his text, but I hope not
essential ones: I hope he will correct me if I show myself to have
misunderstood the nature of his concerns]]
I'm not happy with one aspect of the way he sets up his challenge,
however. He (quite legitimately) demands that whenever we claim to refer
to a particular sequence (as, say, the decimal expansion of some real) we
should know what we are talking about: our ostensible referring expressions
should be well-defined. But in formulating this condition he says:
>Crucial to this assumption is that we have a way of explaining exactly what
>the operation is, i.e. we have a way of putting together a finite sequence
>of English (or French) words so that mathematicians are able to understand
>the meaning of the new name.
-------Now, this seems to me to put altogether too much emphasis on the
expressive resources of existing human languages. (If he had said "we have
a definition of the operation" instead of "we have a way of explaining what
the operation is," I would have been sure he was overemphasizing existing
languages. What he has said is vaguer: I think the appropriate response
depends on just how much leeway we interpret the vagueness as allowing.)
It is surely possible that we can come to understand (and even use in
defining sequences) concepts that cannot be expressed in (current) English
or French. We may (in explaining the concept to an English-speaker) MAKE
USE of English words in teaching someone the new concept, but the English
words will not suffice unaided. As a simple example to get our intuitions
into gear, suppose we discover some novel physical phenomenon
(radioactivity was only discovered a century ago, so the idea of
discovering a novel phenomenon shouldn't seem TOO mysterious). Then we
might explain the concept to someone by saying some words in the presence
of a physical sample: "the average frequency," perhaps, "with which samples
of THAT STUFF emit rays like ... THAT ONE." The English won't help without
the 'ostended' samples.
Now let us consider more mathematical examples. Suppose we already
speak English. (To avoid worries about the "identity criteria" for
languages, suppose we commission the linguists to give us a grammar, and
stipulate that we are interested in the 'fragment' of English defined by
that grammar, with vocabulary drawn from some chosen dictionary. Give the
linguists enough years to fine-tune the grammar and choose a big enough
dictionary and this shouldn't be unduly restrictive.) Is the concept TRUE
SENTENCE OF (stipulated fragment of) ENGLISH one that can be expressed or
explained in (stip...) English?
Certainly, it is not in any straightforward way DEFINABLE in it: that's
what the Paradox of the Liar (or, given our present concern with
enumerating sequences of digits, Richard's Paradox), and the long-running
philosophical debates about its resolution, show!
Is it a concept we can come to have, maybe even one we can help others
come to have? I think so. Certainly the Wittgensteinian troglodyte (excuse
me, that just slipped out: I meant to say "philosophical sceptic") who
persists in saying "But I don't understand" to all our elucidations can't
be PROVEN to be irrational: truth can only be explained to those who "will
not begrudge us some salt." ("Salt," here-- the willingness to make a
philosophical leap of faith/the imagination-- is analogous, I suspect, to
the physical sample of my first example: something necessary that CAN'T be
conveyed through words alone.)
---
If Dean is willing to follow me out on the limb just described, I think
I can suggest something Cantor's argument proves.
Consider a collection of conceptual resources, or, more precisely, a
language. And consider any one-one correspondence between natural numbers
and infinite sequences of digits that can be given in terms of those
resources (i.e. can be uniquely specified-- defined-- in that language).
Then there is a sequence of digits-- the "modified diagonal" of Cantor's
construction-- which is not correlated with any natural number by the given
correspondence. We are saved from contradiction (Richard's Paradox) by the
fact that this sequence may, in particular cases, only be specifiable in
terms of ADDITIONAL conceptual resources (of a more expressive language).
Note that this is a "predicative" version of Cantor's Theorem: so far
no claim has been made about "absolute" infinite cardinalities, and Cantor
is interpreted as having provided a schema by which any GIVEN
correspondence can be shown not to enumerate ALL sequences. The situation
is neatly modeled in Ramified Type Theory ("Principia Mathematica" without
Reducibility). Cantor's argument can be interpreted in this system at EACH
"level" of the ramified hierarchy, and gives a valid proof that the
(level-N) sequences of digits are not correlated with the natural numbers
by any (level-N) one-one correspondence. Fitch ("The hypothesis that
infinite classes are similar," JSL 4 (1939), pp. 159-162), however, showed
that the system could consistently be extended with an axiom scheme saying
that all infinite sets are equinumerous: any (Level:N) infinite set is in a
(level:N+1) one-one correspondence with the natural numbers.
What else does Cantor's argument prove? Well, IF you think you
understand the Platonist's concept of set, then Cantor's argument applies
to the (imaginary and unspeakable and unlearnable-- but that doesn't bother
the Platonist!) "language" with a predicate for every set, and shows that
there is no one-one correspondence WHATSOEVER between the natural numbers
and ALL THE SEQUENCES THERE ARE. But one can go a long way beyond the
scepticism Dean's posting (I may be being unfair here) suggests, and do a
good deal of set theory, without going THAT far with the Platonists (and a
goodly number of foundational researchers in the 20th C were attracted by
such intermediate positions).
---
Allen Hazen
Philosophy Department
University of Melbourne